Open-problem statements, with their sources attached.

We conjecture that the best-known upper bound can be lowered.

We conjecture that the best-known lower bound can be raised.

The analogous problem in $\mathbb{F}^n_p$ remains open. [Gr24]

What is the size of the smallest set $A \subset \mathbb{Z} / p\mathbb{Z}$ (with at least two elements) for which no element in the sumset $A + A$ has a unique representation?

Propose a better lower bound along primes.

Propose a better upper bound along primes.

Can we improve the lower bound $N^{1/2} + O(1)$, at least for infinitely many $N$?

Can we improve the lower bound $N^{1/2} + O(1)$, for all sufficiently large $N$?

Can we improve the upper bound $N^{1/2} + 0.98183 N^{1/4} + O(1)$ [CHO25], at least for infinitely many $N$?

Can we improve the upper bound $N^{1/2} + 0.98183 N^{1/4} + O(1)$ [CHO25], for all sufficiently large $N$?

It is not known whether or not there exists a Sidon subset of $\mathbb{Z}/p\mathbb{Z}$ of size $(1 + o(1))\sqrt{p}$, for all $p$ [Gr24].

It is not known whether, if $G$ is an abelian group of size $n$, there always exists a Sidon subset of $G$ of size $0.01\sqrt{n}$ [Gr24].

Another very nice old problem is whether there is a Sidon subset of $\{0, 1\}^n$ of size $N^{0.51}$, where $N = 2^n$ [Gr24].

Let $p$ be a prime and let $A \subset \mathbb{Z}/p\mathbb{Z}$ be a set of size $\lfloor \sqrt{p} \rfloor$. Is there a dilate of $A$ containing a gap of length $100\sqrt{p}$?

Even what happens in the regime $\omega(p) \sim 10 \log p$ is unclear [Gr24].

Are there infinitely many $q$ for which there is a set $A \subset \mathbb{Z}/q\mathbb{Z}$, $|A| = (\sqrt{2} + o(1))q^{1/2}$, with $A + A = \mathbb{Z}/q\mathbb{Z}$? [Gr24]

Do the following exist, for arbitrarily large $n$? An abelian group $H$ with $|H| = n^{2+o(1)}$, together with subsets $A_1, ..., A_n, B_1, ..., B_n$ satisfying $|A_i||B_i| \ge n^{2-o(1)}$ and $|A_i + B_i| = |A_i||B_i|$, such that the sets $A_i + B_i$ are disjoint from the sets $A_j + B_k$ ($j \neq k$)?

NOTE: according to [CKS05, 4.1], the conditions should be $A_i + B_j$ disjoint from $A_j + B_k$ for $i \neq k$. See green_36.variants.cks05.

Variant using the exact simultaneous double product property from [CKS05, 4.1].

Given a natural number N, what is the smallest size of a subset of ℕ that contains, for each d = 1, …, N, an arithmetic progression of length k with common difference d.

Asymptotic version: determine the asymptotic behavior of m(N, k) as N grows. The solver should determine what function f : ℕ → ℝ eventually equals (fun N ↦ (m N k : ℝ)).